That doesn't look right. As x goes from 0 to 1, ln(1/x) goes from infinity to 0, and ln(ln(1/x)) goes from infinity to -infinity, all monotonically. ln(ln(1/x))=0 at x=1/e and so the integral from 0 to 1/e is real and nonzero, while the integral from 1/e to 1 is imaginary and nonzero, so the answer should be complex.